How to read vertices of quad faces using Python API?
The text editor has a useful template that may interest you:
Text Editor > Templates > Python > BMesh Simple 

Each polygon stores a reference to the indices of the vertices it's made from. 

object mode

import bpy

# obj = bpy.data.objects['Cube']  &lt;-- depending how you want to access it.
obj = bpy.context.active_object

for f in obj.data.polygons:
    for idx in f.vertices:
        print(obj.data.vertices[idx].co)


or manipulate the mesh directly (this will affect all objects that reference this mesh). Meshes and Objects are different things.

import bpy

data = bpy.data.meshes['Cube']

for f in data.polygons:
    for idx in f.vertices:
        print(data.vertices[idx].co)


edit mode

import bpy
import bmesh 

obj = bpy.context.active_object
bm = bmesh.from_edit_mesh(obj.data) 

for f in bm.faces:
    for v in f.verts:
        print(v.co)

# if you make edits then you need to update at the end
bmesh.update_edit_mesh(obj.data)